Measure-theoretic aspects of Convex bodies

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。拟议的研究对象是研究高维凸体或更一般的对数凹概率度量的质量分布。最近，概率论中的几个经典结果被推广到这些度量的更广泛的背景下。这些新的结果似乎超出了经典的基于独立性的概率推理的范围，后者被几何凸性概念所取代。PI打算进一步研究高维测量的几何参数，特别是与这些测量相关的广义形心体的几何。通过这种方法和应用Banach空间局部理论、经典凸性、概率论和信息论的技巧，PI希望解决与对数凹测量有关的各种公开问题，如小球概率估计和超平面猜想，以及凸体理论中的各种猜想，如熵数的正则性和最小平均宽度的最优界。这一提议的研究有一个普遍的原则：高维系统聚集在典型形式周围的趋势。这是一个影响对出现在概率、组合学、统计物理学和复杂性中的复杂系统的研究的中心事实。当高维凸体被看作概率空间时，其出人意料的规律性有望为我们对这一一般原理的理解增加一个新的组成部分。已经发现了随机多面体、随机矩阵和随机算法(分别来自组合学、数学物理和信息学的主题)的应用，这表明未来应该会有更多的应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The object of the proposed research is the study of the distribution of mass of high dimensional convex bodies or more general of log-concave probability measures. Recently, several classical results from probability theory have been extended to the broader setting of these measures. These new results appear to be out of reach of the classical probabilistic reasoning based on independence which is replaced by the geometric notion of convexity. The PI intends to further investigate the geometric parameters of high-dimensional measures and in particular the geometry of generalized centroid bodies associated to these measures. With this approach and applying techniques from local theory of Banach spaces, classical convexity, probability and information theory, the PI wishes to attack various open problems related to log-concave measures such as small ball probability estimates and the hyperplane conjecture, as well as, various conjectures in the theory of convex bodies such as the regularity of the entropy numbers and the optimal bound of the minimal mean width. There is a general principle that underlies the research in this proposal: the tendency of high dimensional systems to congregate around typical forms. This is a central fact that influences the study of complex systems which appear in probability, combinatorics, statistical physics and complexity. The unexpected regularity of high dimensional convex bodies when viewed as probability spaces is expected to add a new component to our understanding of this general principle. Applications to random polytopes, random matrices, and random algorithms (topics from combinatorics, mathematical physics and informatics respectively) have already been discovered, indicating thatmore should be expected in the future.